CorrectionKey=NL-C;CA-C Name Class Date 17 . 1 Angles of ...

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Class

Date

17.1 Angles of Rotation and Radian Measure

Essential Question: What is the relationship between the unit circle and radian measure?

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Explore 1 Drawing Angles of Rotation and Finding Coterminal Angles

In trigonometry, an angle of rotation is an angle formed by the starting and ending positions of a ray that rotates about its endpoint. The angle is in standard position in a coordinate plane when the starting position of the ray, or initial side of the angle, is on the positive x-axis and has its endpoint at the origin. To show the amount and direction of rotation, a curved arrow is drawn to the ending position of the ray, or terminal side of the angle.

In geometry, you were accustomed to working with angles having measures between 0? and 180?. In trigonometry, angles can have measures greater than 180? and even less than 0?. To see why, think in terms of revolutions, or complete circular motions. Let be an angle of rotation in standard position.

terminal side

y

x

initial side

? If the rotation for an angle is less than 1 revolution in a counterclockwise direction, then the measure of is between 0? and 360?. An angle of rotation measured clockwise from standard position has a negative angle measure. Coterminal angles are angles that share the same terminal side. For example, the angles with measures of 257? and -103? are coterminal, as shown.

y

x

257?

-103?

? Houghton Mifflin Harcourt Publishing Company

? If the rotation for is more than 1 revolution but less than 2 revolutions in a counterclockwise direction, then the measure of is between 360? and 720?, as shown. Because you can have any number of revolutions with an angle of rotation, there is a counterclockwise angle of rotation corresponding to any positive real number and a clockwise angle of rotation corresponding to any negative real number.

Module 17

825

y x

420?

Lesson 1

A Draw an angle of rotation of 310?. In what quadrant is the terminal side of the angle?

y x

B On the same graph from the previous step, draw a positive coterminal angle. What is the

angle measure of your angle? y x

C On the same graph from the previous two steps, draw a negative coterminal angle. What is

the angle measure of your angle? y x

? Houghton Mifflin Harcourt Publishing Company

Reflect

1. Is the measure of an angle of rotation in standard position completely determined by the position of its terminal side? Explain.

2. Find the measure between 720? and 1080? of an angle that is coterminal with an angle that has a measure of -30?. In addition, describe a general method for finding the measure of any angle that is coterminal with a given angle.

Module 17

826

Lesson 1

Explore 2 Understanding Radian Measure

The diagram shows three circles centered at the origin. The arcs that are on the circle between the initial and terminal sides of the 225? central angle are called intercepted arcs. ABis on a circle with radius 1 unit. CDis on a circle with radius 2 units. EFis on a circle with radius 3 units.

y 3

2 1 225?

B D F

Notice that the intercepted arcs have different lengths, although they are intercepted by the same central angle of 225?. You will now explore how these arc lengths are related to the angle.

AC E x

The angle of rotation is degrees counterclockwise.

There are degrees in a circle.

225? represents _ ____of the total number of degrees in a circle.

So, the length of each intercepted arc is _ ____of the total circumference of the circle that it lies on.

B Complete the table. To find the length of the intercepted arc, use the fraction you found in

the previous step. Give all answers in terms of .

Radius, r 1

Circumference, C

( C = 2r)

__ Length of

Ratio of Arc Length to

Intercepted Arc, s

Radius, rs

2

3

Reflect

3. What do you notice about the ratios _rs in the fourth column of the table?

4. When the ratios of the values of a variable y to the corresponding values of another variable x all equal

Ba eccoanusstean_xy_t=k,

y is said to k, you can

be proportional to solve for y to get y

x, and = kx.

the constant k is called the constant of proportionality. In the case of the arcs that are intercepted by a

225? angle, is the arc length s proportional to the radius r? If so, what is the constant of proportionality,

and what equation gives s in terms of r?

5. Suppose that the central angle is 270? instead of 225?. Would the arc length s still be proportional to the radius r? If so, would the constant of proportionality still be the same? Explain.

? Houghton Mifflin Harcourt Publishing Company

Module 17

827

Lesson 1

Explain 1 Converting Between Degree Measure and

Radian Measure

y

For a central angle that intercepts an arc of length s on a circle with radius r, the radian measure of the angle is the ratio = _rs. In particular, on a unit circle, a circle centered at the origin with a radius of 1 unit, = s. So, 1 radian is the angle that intercepts an arc of length 1 on a unit circle, as shown.

s 1

2x 1

Recall that there are 360? in a full circle. Since the circumference of a circle of

radius

r

is

s

=

2r,

the

number

of

radians

in

a

full

circle

is

_2__r r

=

2.

Therefore,

360?

=

2

radians.

So,

1?

=

_2__ 360

=

___ 180

radians

and

1

radian

=

_3_60_ 2

=

_1_80_

degrees.

This result is summed up in the following table.

= 1 radian

CONVERTING DEGREES TO RADIANS

( ) Multiply the number of degrees by

__r_a_d_ia_n_s 180?

.

CONVERTING RADIANS TO DEGREES

( ) Multiply the number of radians by

__1_8_0_?__ radians

.

Example 1 Convert each measure from degrees to radians or from radians to degrees.

A Degree measure

20?

Radian measure

____ 180?

?

20?

=

__ 9

315?

____ 180?

?

315?

=

_7__ 4

600?

____ 180?

?

600?

=

_1_0__ 3

-60?

____ 180?

?

(-60?)

=

-_3_

-540?

____ 180?

?

(-540?)

=

-3

B Radian measure

__ 8

_4__ 3

_9__ 2

-_71_2_

-_1_36__

Degree measure

_1_8_0_?

?

__ 8

=

?

_4__ 3

=

? Houghton Mifflin Harcourt Publishing Company

Module 17

828

Lesson 1

Reflect

6. Which is larger, a degree or a radian? Explain.

7. The unit circle below shows the measures of angles of rotation that are commonly used in trigonometry, with radian measures outside the circle and degree measures inside the circle. Provide the missing measures.

y

2

3

4

120?

60?

4

150?

30?

x

0

210?

330?

5

240?

4

300? 7 4

3 2

Your Turn

Convert each measure from degrees to radians or from radians to degrees. 8. -495?

9. _ 1132

? Houghton Mifflin Harcourt Publishing Company

Module 17

829

Lesson 1

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